A new investigation on fractionalized modeling of human liver

This study focuses on improving the accuracy of assessing liver damage and early detection for improved treatment strategies. In this study, we examine the human liver using a modified Atangana-Baleanu fractional derivative based on the mathematical model to understand and predict the behavior of the human liver. The iteration method and fixed-point theory are used to investigate the presence of a unique solution in the new model. Furthermore, the homotopy analysis transform method, whose convergence is also examined, implements the mathematical model. Finally, numerical testing is performed to demonstrate the findings better. According to real clinical data comparison, the new fractional model outperforms the classical integer-order model with coherent temporal derivatives.


Essential definitions
Some of the major mathematical definitions utilised in this work are as follows: Definition 2. 1 The following Mittag-Leffler functions, E ρ (G ) and E ρ,τ (G ) , were introduced by Mittag-Leffler 22 and Wiman 23 , respectively.and Definition 2. 2 The standard definition of the Riemann-Liouville fractional integral 24 of h of order ℜ(ν) > 0 is given as follows: Definition 2. 3 Let g be a real-valued picewise continuous function (0, ∞) .The Laplace transform (LT) of g(z) 25 of exponential order α > 0 concerning z is given as follows; Definition 2. 4 For the function ḡ(s) , the inverse Laplace transform with respect to y ≥ 0 is given as follows 25 ; here Γ ∈ R is a constant.Definition 2. 5 The standard definition of the MABC 10 of f of order ϑ ∈ (0, 1) is given as follows: where µ ϑ = ϑ 1−ϑ , and M(0) = 1 = M(1).
Definition 2. 6 The novel fractional derivative with a non-local kernel associated with the MAB fractional integral 10 is defined as follows:

Formulation of the model
Čelechovská 17 proposed a model of the human liver with integer order in 2004.For parameter identification within that study, the author employed the clinical information collected by the BSP test.Let R(t) and W(t) represent the concentrations of BSP in blood and liver at time t, respectively; the flow transfer of BSP between them is shown in Figure 1.Then, the integer-order model suggested by Čelechovská is described as follows: Where R(0) = R 0 and W(0) = W 0 are initial conditions ( R 0 > 0 ), and the transfer rates are represented by known parameters α, β, δ.
We stabilize the system by replacing the time derivative with the MABC 10 .The right, as well as the left sides, will no longer be the same dimension as with this alteration.To resolve this issue, we adjust the fractional operator such that both sides have an equal dimension by using an additional parameter with the dimension of time called ρ 26 .The reasoning provided leads to the following explanation of the Human liver fractional model for t ≥ 0 and ϑ ∈ (0, 1) The next section investigates whether the system (2) solution to the FPT exists and is unique.

Presence of a unique solution
In this part, we demonstrate that the system has a singular solution.To do this, we use Nieto and Losada's 27 fractional integral operator on the system (2), and we get Using the definition of MABFI 10 , we have For convenience, we consider MAB 0 . (1) )], The Lipschitz condition (LC) and contraction are fulfilled by the kernel q 1 if the following disparity persists: 0 < α ≤ 1.
Proof Consider the function R(t) and R 1 (t) , then Thus, the LC is fulfilled for q 1 .Additionally, if 0 < α ≤ 1 , then q 1 is a contraction.
Similarly, q 2 satisfy the LC as follows: With q 1 , q 2 , in mind, the Eq. ( 3) may be written as follows: Thus, consider the below recursive formula: where R 0 (t) = R(0), W 0 (t) = W(0).Now, we examine and Using the aforementioned equations, one may write Using the triangular inequality and the L 1n definition, we obtain Vol The aforementioned result demonstrates that the system (2) has a solution.
Proof Assume that R(t), W(t) are bounded function.We have demonstrated that kernels L in , i = 1, 2 satisfy the LC.Using the outcome of ( 5) and ( 6) together with the recursive technique, we obtain As a result, smooth functions (4) exist.We assert that the aforementioned operations are the system's (2) solutions.To back up this assertion, we suppose

We get
We get by continuing this process Using the recent equation's limit as n → ∞ , we obtain �X 1n (t)� → 0 .In the same manner, we obtain �X 2n (t)� → 0 and this proof is finished.
To demonstrate the solution's uniqueness, we presume that the system (2) has an alternative solution, such as R 1 , W 1 .Then We obtain, according to the LC of R Thus Theorem 4. 3 The Human-Liver model (2) solution is unique if the following requirements are satisfied: Proof We may deduce from the condition (8) and Eq. ( 7) that So �R(t) − R 1 (t)� = 0 , then R(t) = R 1 (t) .Similarly, we may demonstrate that W(t) = W 1 (t).
The proof is finished.

Stability analysis by FPT
We find a particular solution to the Human-Liver model using the Laplace transform, and then we use FPT to demonstrate the iterative method's stability.We begin by utilizing the LT to both sides of the equations in the model ( 2), then We conclude from the LT definition of the MABC the following: If we reorder the inequalities shown above, then

Evaluation of the iteration method's stability
Let us consider the recursive method p n+1 = χ(T , R n ) and a self-map T on Banach space (H, .) .Presume that ζ(T ) = φ is the fixed point set of T and which is lim n→∞ p n = p ∈ ζ(T ) .Presume that {t n } ⊂ ζ and j n = �t n+1 − χ(T , t n )� .If lim n→∞ j n = 0 =⇒ lim n→∞ t n = p , then the iterative process p n+1 = χ(T , p n ) is T-stable.Presume there is an upper limit for our sequence {t n } .If all of these circumstances are fulfilled for Picard's iteration p n+1 = T p n , then p n+1 = T p n is T-stable.
Theorem 5.1 Wang et al. 28 Let T be a self-map on Banach space (H, .) , satisf ying �T y − T z � ≤ A�y − T y � + a�y − z� for all y, z ∈ H where A ≥ 0 and 0 ≤ a < 1 .Assume that T is Picard T-stable.
As per Eq. ( 9), the Human-Liver (2) fractional model is related to the subsequent iterative formula.Now take a look at the following theorem.

Theorem 5.2 Assume that T is a self-map defined as follows:
If the following conditions are met, this iterative recursive is T-stable in L 1 (a, b): Proof To demonstrate that T has a fixed point, For (k, l) ∈ N × N , we compute the following inequalities: Applying the norm on both halves, we get Since the roles in the solution are the same, we may think about From Eqs. (10) and (11), we get  12), ( 13) hold, we suppose A = (0, 0), Consequently, all of Theorem (5.1)'s conditions have been fulfilled, and the proof is finished.

Numerical implementation
In this part, we apply HATM (homotopy analysis transform method) to build the model ( 2) reasonably accurately.
It should be noted that HATM is a well-developed improvement of the traditional LT method 29 and HAM 30 .To keep the homogeneity and stability assessment of the proposed model accurate and precise, the stability and structure of the model equations under different conditions have been thoroughly considered.To solve the model (2) using HATM, we first apply LT as shown below: which results in Then we have Using the homotopy method, we define Consequently, the deformation equations become where r ∈ [0, 1] denotes an embedding parameter, Φ j (t; r), j = 0, 1 are unknown functions, R 0 , W 0 are pre- liminary estimates, h = 0 is an auxiliary parameter, L [.] is the Laplace operator, and H(t) = 0 is an auxiliary function.
Here, we expand Φ j (t; r) (j = 1, 2) in the Taylor series concerning r.This technique produces where According to Liao 30 , the series (15) converges at r = 1 if H(t), h, and the starting estimates are suitably selected.Thus, we obtain Additionally, the η th order deformation equation may be expressed as where and Using the inverse LT to Eq. ( 16), we obtain Through the solution of these equations for various values of η = 1, 2, 3, • • • , we get to Ultimately, the outcomes of system (2) are achieved as follows: and Derivative of R(t) and W(t) (with H = 1, t = 0, and ϑ = 1 ) as follows: and

Convergency of HATM of FDEs
In this part, we examine the convergence of HATM by introducing and demonstrating the following theorem.

Discussion and results
This part presents the HATM to give a numerical simulation for the Human-Liver model (2).The BSP model is solved using parameters selected from Table 1 for the computational simulation.
In the beginning, the effect of h on the convergence of the series solutions derived by the HATM is explored.This parameter sets the convergence zone and rate of convergence for the HATM approximations.We depict the h-curves for the approximate solutions to Eq. ( 2) to understand the influence of h.
Figure 2 shows plots of h with R ′ (0), R ′′ (0), R ′′′ (0) and W ′ (0), W ′′ (0), W ′′′ (0) for ϑ = 1 .Based on the figure, we pick the horizontal line parallel to the h-axis as the convergence zone for the approximations, therefore the convergence of the approach is assured for −1.5 ≤ h ≤ −0.5 .As a result, the middle point of this range, h = −1 , is an excellent choice for h at which the numerical solution converges 31 .
The Adomian Decomposition Method (ADM) 32 , the CFFD 20 , and the MABC findings are compared in Tables 2 and 3 using clinical data collected by Evzen Hrncif in 1985 17 with h = −1, ϑ = 1 and H = 1 .As shown, the Table 1.The variables and parameter values of the model (Figure 1) 17,20 .

R(t)
The concentration of BSP in the blood at any time t R(0) = 250 The findings are shown in Figure 3, demonstrating that as ϑ → 1 , the approximate solutions converge to the traditional integer answers.The MABC also performs well for ϑ 's around 1 and lesser ones.Figure 4 shows the measured values of BSP in blood and bile, corresponding to different values of the unknown parameters α, β , and δ.

Conclusion
In this study, we propose a novel MABC fractional derivative model of the human liver using a nonsingular kernel that not only contributes to its mathematical robustness but also aligns with the biological complexity of the human liver.A parameter adjustment has been used to prevent dimensional mismatching.In addition, the iteration method and fixed point theory were used to look into the possibility of a single solution.A fresh and potent numerical approach was also offered to execute the proposed model in an acceptable, accurate way.The recommended method's stability was also examined and assessed.Additionally, certain numerical tests in Tables 2  and 3 and Figure 3 were used to confirm the effectiveness of the new strategy.Figure 3 also demonstrates the viability of the suggested fractional calculus modeling.Compared with earlier studies, the specific order of the novel fractional models follows reality better than the classic integer solutions.This study may not be entirely responsible for diverse biological characteristics between different patients.The model's projection may be limited for a broader population, and its forecasting accuracy may vary in individuals.For example, according to the perspective of quantifying BSP, Figure 4 is related to various values of unknown factors α, β , and δ .These parameters may be crucial in the pharmaceutical sector.According to this study, the estimate of human liver damage is known earlier than the integer order, which is necessary for much better treatment.Future research should focus on refining the model by incorporating more biological realism, considering personal variability, and conducting complete verification in diverse clinical scenarios.For various values of α, β , and δ , the graphs of R(t) and W(t) are (a) and (b), (c) and (d), and (e), respectively.

Figure 1 .
Figure 1.Simple process of BSP administration through blood and liver.
MABC offers more accurate findings than the ADM and HATM in terms of actual experimental observations.The MABC solutions are then calculated for values of ϑ = 0.610, 0.615, 0.620, 0.625, 0.630, 0.635, h = −1, and H = 1 .